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<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><b>suppose </b></a><a NAME="E1:21_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:21_1_1"/>
( <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T56">NAT_1:56</a></i>;<br/><a NAME="E3:21_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">k'</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:21_1_1"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k'</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>now </b></a><div class="add"><a NAME="E1:21_1_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E2:21_1_1_1"/>
<font color="Maroon">k'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:21_1_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:21_1_1_1"/>
<font color="Maroon">k'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>assume </b><a NAME="E6:21_1_1_1"/>
( <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> &amp; <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> )
 ;<br/><a NAME="E7:21_1_1_1"/><b>then </b>
( <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">p'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">p'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/><a NAME="E8:21_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k'</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><a NAME="E9:21_1_1_1"/><b>then </b>
<font color="Maroon">p'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k'</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E10:21_1_1_1"/>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k'</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:21_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2"><b>suppose </b></a><a NAME="E1:21_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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